Algebraic families of Galois representations and potentially semi-stable pseudodeformation rings

Abstract

We construct and study the moduli of continuous representations of a profinite group with integral $p$-adic coefficients. We present this moduli space over the moduli space of continuous pseudorepresentations and show that this morphism is algebraizable. When this profinite group is the absolute Galois group of a $p$-adic local field, we show that these moduli spaces admit Zariski-closed loci cutting out Galois representations that are potentially semi-stable with bounded Hodge-Tate weights and a given Hodge and Galois type. As a consequence, we show that these loci descend to the universal deformation ring of the corresponding pseudorepresentation.

Numbering changed and typos corrected to match published version. 59 pages

References in corpus (3)

• Local models for Weil-restricted groups
• Pseudo-modularity and Iwasawa theory
• Ordinary pseudorepresentations and modular forms

Cited by in corpus (12)

• Coherent sheaves on the stack of Langlands parameters
• Pseudo-modularity and Iwasawa theory
• The rank of Mazur's Eisenstein ideal
• Ordinary pseudorepresentations and modular forms
• On local Galois deformation rings
• Class groups and local indecomposability for non-CM forms
• Deformation conditions for pseudorepresentations
• On the automorphy of 2-dimensional potentially semi-stable deformation rings of $G_{{\mathbb{Q}}_p}$
• Rigidity of Automorphic Galois Representations over CM Fields
• Finiteness properties of the category of mod $p$ representations of ${\mathrm{GL}}_2({\mathbb{Q}}_p)$
• Cohomology of $(\phi ,\Gamma )$-modules over pseudorigid spaces
• The Chabauty-Kim Method for Relative Completions